2018/01/18 by Kirti Joshi, Joshi, Kirti
Computer Science · Mathematics · #11G05 #14G05 #14H52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11G05 #msc:14G05 #msc:14H52
paper · pdf · doi:10.48550/arxiv.1801.06245
Five pages. Edits to Introduction
openalex publication_date 2018/01/18 · arxiv created 2019/10/15 · arxiv updated 2019/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational function field in characteristic p>0. In particular I show that if n≥ 4 is any even integer and not divisible by the characteristic of the field then any elliptic curve E over this field has at least 2n rational points over a finite solvable field extension. Under additional hypothesis, when the ground field is a number field, I show that these are of infinite order. I also show that Ulmer's points lift to characteristic zero and in particular to the canonical lifting.